Full runner removal theorem for Ariki-Koike algebras

Abstract

We consider the representation theory of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r \wr S_n$. We define the addition of a runner full of beads for the abacus display of a multipartition and investigate some combinatorial properties of this operation. We focus our attention on the $q$-decomposition numbers, i.e. the polynomials arising from the Fock space representation of the quantum group $U_q(\widehat{\mathfrak{sl}}_e)$. Using Fayers' LLT-type algorithm for Ariki-Koike algebras, we relate $q$-decomposition numbers for different values of $e$ for the class of $e$-multiregular multipartitions, by adding a full runner of beads to each component of the abacus displays for the labelling multipartitions.